Problem 2 - Applied Calculus

Question

The population of a city, \(P\), in millions, is a function of \(t\), the number of years since 1970 , so \(P=f(t)\). Explain the meaning of the statement \(f(35)=12\) in terms of the population of this city.

Step-by-Step Solution

Answer

In 2005, the city's population was 12 million.

Step 1: Identify Given Information

We are given that the function \(f(t)\) represents the population \(P\) of a city in terms of \(t\), the number of years since 1970. The statement \(f(35)=12\) is provided.

Step 2: Understand the Variables

In the function \(f(t)\), \(t\) is the number of years since 1970. This means that \(t=0\) corresponds to the year 1970. Therefore, \(t=35\) corresponds to the year 2005.

Step 3: Interpret the Given Equation

The statement \(f(35)=12\) indicates that when \(t=35\), the population \(P\) is 12 million. Since \(t=35\) corresponds to the year 2005, this means the population in 2005 was 12 million.

Key Concepts

Population Growth through Functions

When we talk about population growth in the context of calculus and functions, we describe how the population of a place, such as a city, changes over time. A typical way to represent this is through a mathematical function, often noted as \( P = f(t) \). Here, \( P \) is the population, usually given in millions for large areas, and \( t \) is the time that has passed since a starting point—in this case, the year 1970. This approach allows us to see how certain mathematical expressions predict the growth or decline of a population.

These functions are crucial for understanding demographic trends. Governments and planners rely on them to make decisions about infrastructure and resource allocation. Knowing how to interpret a function is essential because it provides a snapshot of population dynamics over time. By substituting different values of \( t \) into the function, it shows how the population changes over specific time intervals.

For instance, if a city planner sees that \( f(35) = 12 \), they can deduce that in 2005 (35 years after 1970), the population was 12 million. This understanding helps inform policies related to growth, sustainability, and urban development.

Interpreting Mathematical Functions

Interpreting mathematical functions involves understanding what specific components of the function represent in real-world terms. A function like \( f(t) \) not only provides an abstract representation but conveys concrete information through values and expressions.

To interpret \( f(35) = 12 \), we break it down:

• Identify the function: \( f(t) \) denotes the population depending on time.
• Understand variables: \( t \) stands for the number of years since a baseline year (1970).
• Read the function: \( f(35) \) implies the population at a specific year, 35 years post-1970.
• Extract the meaning: The function's result, 12, means the population in 2005 was 12 million.

Functions serve as powerful tools in calculus that help convert numerical data into meaningful interpretations. Such interpretations enlighten us about trends and forecasts, guiding effective decision-making.

Understanding Time-Dependent Variables

Time-dependent variables in functions capture how a phenomena, such as population, evolves as time progresses. This concept is pivotal in calculus when modeling real-world situations that change over time.

In our example, \( t \) is the time-dependent variable, representing the years that have passed since 1970. As \( t \) increases, we aim to determine the changes in another variable, \( P \), the population, which likewise evolves.

This dynamic can be visualized as a curve on a graph, where the y-axis represents population, and the x-axis represents time.

• Each point on the curve corresponds to a specific time (\( t \)) and its respective population (\( P \)).
• Understanding this relationship helps in forecasting future trends and planning.
• The real-world applicability of such models is vast, including ecological studies, economics, and public health planning.

Time-dependent variables give structure to our understanding of how the world operates over time, reinforcing the utility of mathematical functions in analytical predictions.